In this work, a kernel-based Ensemble Gaussian Mixture Probability Hypothesis Density (EnGM-PHD) filter is presented for multi-target filtering applications. The EnGMPHD filter combines the Gaussian-mixture-based techniques of the Gaussian Mixture Probability Hypothesis Density (GMPHD) filter with the particle-based techniques of the Sequential Monte Carlo Probability Hypothesis Density (SMC-PHD) filter. It achieves this by obtaining particles from the posterior intensity function, propagating them through the system dynamics, and then using Kernel Density Estimation (KDE) techniques to approximate the Gaussian mixture of the prior intensity function. This approach guarantees convergence to the true intensity function in the limit of the number of components. Moreover, in the special case of a single target with no births, deaths, clutter, and perfect detection probability, the EnGM-PHD filter reduces to the standard Ensemble Gaussian Mixture Filter (EnGMF). In the presented experiment, the results indicate that the EnGMPHD filter achieves better multi-target filtering performance than both the GM-PHD and SMC-PHD filters while using the same number of components or particles.

The solution to the state estimation problem is given by the Bayesian recursive relations (BRRs). Recently, ensemble Gaussian mixture filters have shown to be an accurate and consistent solution to the state estimation problem. In this type of filters, the BRRs are solved by approximating the state probability density function (PDF) via Gaussian mixtures (GMs) and point masses (PMs). Throughout the propagation and measurement update steps, the approximated state PDF is constantly switching between GMs and PMs. Therefore, a key step for this solution involves optimally sampling PMs from GMs. For onboard applications, verifiable and computationally inexpensive sampling techniques are crucial. In previous work, a deterministic sampling technique was developed by minimizing a distance metric known as the modified Cramér-von Mises distance (MCVMD), yielding a verifiable solution. However, the computationally feasibility of this solution for onboard use was not considered. This work introduces a new sampling strategy that is both deterministic and computationally inexpensive compared to MCVMD approach. By solving the approximate optimal transport problem via an iterative Sinkhorn-Knopp algorithm, this new technique is able to sub-optimally sample from a GM, providing a computationally inexpensive filter.

Gaussian mixture-type filters have become indispensable tools for modeling intricate and nonlinear systems, offering a departure from traditional Gaussian-centric approaches. This work focuses on the critical aspect of accurate weight computation during the measurement incorporation phase of Gaussian mixture filters. The proposed novel approach computes weights by linearizing the measurement model about each component’s posterior estimate rather than the the prior, as traditionally done. This work proves equivalence with traditional methods in linear scenarios and empirically demonstrates improved performance in nonlinear cases. Two illustrative examples, the Avocado and Lorenz’ 63 models, serve to elucidate the advantages of the new weight computation technique by analyzing filter accuracy and efficiency through varying the number of Gaussian mixture components.

The accuracy of the point mass filter (PMF) relies on the precise placement of grid points. Since the approximated probability distributions are evaluated only at these points, suboptimal choices in grid placement can result in an inaccurate representation of the posterior distribution. This work addresses this issue by representing the propagated grid points as a Gaussian mixture, enabling a Gaussian sum filter (GSF) update before grid construction. The use of the GSF update enhances the accuracy of the mean and covariance estimates, from which a new grid can be constructed. This approach leads to improved grid placement and reduces the number of points required to achieve satisfactory results. A comparative analysis is conducted between this new approach, the traditional PMF, and a PMF variant that uses an unscented Kalman filter update before grid construction. Using a simple bivariate example, the new variant is shown to approximate the posterior distribution better than the other filters. Furthermore, the new approach is evaluated in two sequential filtering problems: the first involves the Ikeda map, and the second focuses on terrain-relative navigation for Martian exploration. The results show a more accurate, and more consistent filter compared to the other two PMF variants considered.

The Burnished Flow Filter is a particle flow filter constructed from the Kalman filter measurement update equations. The derivation for this filter begins by assuming the classic Kalman Filter measurement update equations are the solution to a stochastic differential equation. By using these well known equations, the derivation of this filter follows naturally to an engineer with a Kalman filtering background. The work presented here shows the derivation, and application of this filter on both linear and nonlinear problems. The Burnished Flow Filter is benchmarked against the widely used Gromov Flow Filter, revealing similar performance in linear problems and demonstrating superior consistency in the nonlinear scenarios under study. Additionally, the Burnished Flow Filter exhibits a smoother flow compared to the Gromov Flow Filter, as evidenced by a smaller state update during the first substep of the measurement update.

The incorporation of nonlinear measurement information plays an important role in Bayesian state estimation for real-word systems. While many methods exist for propagating states through continuous-time nonlinear dynamics, a complementary continuous solution for discrete-time nonlinear measurements has so far remained elusive. Building on intuition from our previous work, the Bayesian Recursive Update Filter, we formulate the nonlinear measurement update as an ordinary differential equation (ODE). This formulation naturally extends to particle flow. We define two particle flows: the first is a deterministic flow based on the ODE solution, and the second is stochastic; the numerical integration contains a diffusion term. The proposed particle flows demonstrate excellent performance on a system with deterministic dynamics and a highly accurate nonlinear measurement, a setting known to be challenging for particle filters.

In the high-dimensional setting, Gaussian mixture kernel density estimates become increasingly suboptimal. In this work we aim to show that it is practical to instead use the optimal multivariate Epanechnikov kernel. We make use of this optimal Epanechnikov mixture kernel density estimate for the sequential filtering scenario through what we term the ensemble Epanechnikov mixture filter (EnEMF). We provide a practical implementation of the EnEMF that is as cost efficient as the comparable ensemble Gaussian mixture filter. We then showcase that the EnEMF has a significant reduction in error per particle on the 40 -variable Lorenz ‘ 96 system. We answer the titular question, “are non-Gaussian kernels suitable for ensemble mixture model filtering?” in the affirmative.

Nonlinear measurement models pose a challenge to linear filters. The ensemble Kalman filter (EnKF) is a popular choice despite its tendency to diverge in systems with highly accurate, highly nonlinear measurements. In this work, we present the Bayesian Recursive Update EnKF (BRUEnKF): a novel EnKF that employs the Bayesian Recursive Update Filter (BRUF) measurement update. The BRUF divides the the extended Kalman filter (EKF) update into an integer number of steps, allowing for the recomputation of the measurement Jacobian at regular intervals. We adapt the BRUF update for an ensemble filter, taking advantage of the EnKF’s numerical covariance computation at each update step. The BRUEnKF is shown to outperform the EnKF for systems with range measurements.

The ensemble Gaussian mixture filter (EnGMF) is a powerful filter for highly non-Gaussian and non-linear models that has practical utility in the case of a small number of samples, and theoretical convergence to full Bayesian inference in the ensemble limit. We aim to increase the utility of the EnGMF by introducing a particle-local notion of covariance into the Gaussian mixture estimate of the prior distribution. We show on a simple bivariate problem that each particle having its own local estimate of the covariance both has nice qualitative and quantitative properties, and significantly improves the estimate of the prior and posterior distributions for all ensemble sizes. We additionally show the utility of the proposed methodology for sequential filtering for the Lorenz ’63 equations, achieving a significant reduction in error in the low ensemble size regime.

The proliferation of large satellite constellations in low Earth orbit (LEO) is dramatically increasing demand on existing systems for space domain awareness. The rapidly growing number of objects in LEO will reduce the average rate of observations per object, necessitating the development of multi-target algorithms that can handle higher levels of data sparsity without sacrificing computational efficiency. In this paper, we demonstrate that a multi-target filter that combines a number of useful capabilities is able to track and maintain custody of a simulated population of over 16,000 LEO objects without requiring the use of high-performance computing facilities. The filter is based on the generalized labeled multi-Bernoulli filter, and includes three previously-presented features: label space partitioning based on sensor fields of view, the ensemble Gaussian mixture filter (EnGMF), and bi-fidelity orbit uncertainty propagation. We also introduce a new, algorithmically simple method for adjusting the number of EnGMF particles to balance accuracy and computational efficiency, which we refer to as progressive resampling.

Filters relying on the Gaussian approximation typically incorporate the measurement linearly, i.e., the value of the measurement is pre-multiplied by a matrix-valued gain in the state update. Nonlinear filters that relax the Gaussian assumption, on the other hand, typically approximate the distribution of the state with a finite sum of point masses or Gaussian distributions. In this work, the distribution of the state is approximated by a polynomial transformation of a Gaussian distribution, allowing for all moments, central and raw, to be rapidly computed in closed form. Knowledge of the higher-order moments is then employed to perform a polynomial measurement update, i.e., the value of the measurement enters the update function as a polynomial of arbitrary order. A filter employing a Gaussian approximation with linear update is, therefore, a special case of the proposed algorithm when the order of the update is set to one. At the cost of more computations, the new methodology guarantees performance better than the linear/Gaussian approach for nonlinear systems. This work employs monomial basis functions and Taylor series, but it is readily extendable to an orthogonal polynomial basis.

A novel adaptive nonlinear estimator is presented to accurately incorporate nonlinear/non-Gaussian measurement in a Bayesian framework. The underlying algorithm relies on a Gaussian Mixture Model (GMM) to approximate the probability density function (pdf) of the state conditioned on all current and past measurement. Automatic mixture components refining is performed to ensure that the posterior GMM approximation of the pdf accurately represents the true distribution.